Hello everyone,
One of the schools I am applying to asks for a "personal statement of teaching and research philosophies."
Is this a fancy way of saying we want a teaching statement and a separate research statement?

I agree with above: if they want them separate, they are usually listed in lists, separated by commas or even numbered or bulleted. I read this as, "give us one document", even if it is a vague way to say it.

What does the "personal" part refer to? Does that mean they want a narrative or are they just saying tell us what YOU think.
Any suggestions on how do you write a single statement on teaching and research? What if your research has absolutely nothing to do with your teaching experience or teaching philosophy

"Any suggestions on how do you write a single statement on teaching and research?"
You could just have it be one document with two sections.

"What if your research has absolutely nothing to do with your teaching experience or teaching philosophy."
In my experience, the schools that ask for this are a bit more teaching-oriented than average. So they do prefer the research and teaching to be linked in some way. Think about the teaching of research, for example.

I'll preface this by saying that I in no way claim that what I did is right, BUT, when it was somewhat ambiguous I always just sent two separate statements. In fact, I never sent one single statement of research and teaching. I always just assumed they worded the ad poorly, and (like Anon1983) I cannot wrap my head around how to write one cohesive statement on both.

I will also mention that for one job that had an earlier deadline last year I emailed the search chair and asked about this very issue. S/he said that sending either one or two is fine. So I used that as justification for my actions, however meager that justification may be. So perhaps that is the answer — if the ad seems unclear to you, just send a quick email and ask. The worst they can do is not respond.